Complete minimal surfaces densely lying in arbitrary domains of ℝn

Alarcón, Antonio; Castro-Infantes, Ildefonso

doi:10.2140/gt.2018.22.571

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Complete minimal surfaces densely lying in arbitrary domains of $\mathbb{R}^n$

Antonio Alarcón and Ildefonso Castro-Infantes

Geometry & Topology 22 (2018) 571–590

DOI: 10.2140/gt.2018.22.571

Abstract

In this paper we prove that, given an open Riemann surface $M$ and an integer $n \geq 3$ , the set of complete conformal minimal immersions $M \to ℝ^{n}$ with $\bar{X (M)} = ℝ^{n}$ forms a dense subset in the space of all conformal minimal immersions $M \to ℝ^{n}$ endowed with the compact-open topology. Moreover, we show that every domain in $ℝ^{n}$ contains complete minimal surfaces which are dense on it and have arbitrary orientable topology (possibly infinite); we also provide such surfaces whose complex structure is any given bordered Riemann surface.

Our method of proof can be adapted to give analogous results for nonorientable minimal surfaces in $ℝ^{n} (n \geq 3)$ , complex curves in $ℂ^{n} (n \geq 2)$ , holomorphic null curves in $ℂ^{n} (n \geq 3)$ , and holomorphic Legendrian curves in $ℂ^{2 n + 1} (n \in ℕ)$ .

Keywords

complete minimal surface, Riemann surface, holomorphic curve

Mathematical Subject Classification 2010

Primary: 49Q05

Secondary: 32H02

References

Publication

Received: 15 November 2016

Accepted: 28 April 2017

Published: 31 October 2017

Proposed: Tobias H Colding

Seconded: Dmitri Burago, Leonid Polterovich

Authors

Antonio Alarcón
Departamento de Geometría y Topología Universidad de Granada Granada Spain
http://www.ugr.es/~alarcon
Ildefonso Castro-Infantes
Departamento de Geometría y Topología Universidad de Granada Granada Spain
http://www.ugr.es/~icastroinfantes

Volume 22, issue 1 (2018)